How does a tv change a video signal into a picture?

This is way simplified.

First - do you understand raster : the fact that the screen that you're looking at right now consists of about a thousand horizontal lines ?

If so, the theory is easy -

as the signal comes in the amplitude of that signal is translated into brightness of the current point on the line (for color - you have 3 brightnesses [red/blue/green])

The point moves back and forth, down and up, to cover the whole screen 30 times a second.